First of all, if you haven't seen the series finale, this answer might be considered a spoiler (it was for me, as I haven't seen the finale yet). But I'm also not going to put the whole answer into a spoiler block, so if you have fear DON'T READ ANY FURTHER.
I found this very interesting article about exactly this question. It argues that How I Met Your Mother is basically a prequel to Full House and Ted Mosby (voiced in part by Bob Saget) is Danny Tanner (played by Bob Saget). Now of course it is clearly not a direct prequel, given the different persons, setting and timeframe, but maybe a prequel in spirit.
As we learn in the series finale, the Mother is already dead for six years when Ted tells his overlong story to his children and this ending was planned this way right when the show started (as also confirmed in the answers to this related question):
Carter Bays has confirmed that the series' tragic close was "the
ending they conceived when they conceived the show 10 years ago."
Having envisioned a show all about a man who would ultimately lose his
wife telling his children how he met her, Bays and Thomas cast Bob
Saget in the part - the most famous widower in modern sitcoms.
Furthemore, the article notes some similarities between Ted's group and the characters from Full House:
Ted, as played by Josh Radnor, is Danny Tanner, just with a better sex
life and an all-consuming love of architecture. The inherent, series
spanning joke about both men is that they are nerdy, 'dad joke' making
father figures. They're both the same archetype within their
respective series - 'the dad'. Bays and Thomas wrote a part strikingly
similar to Danny Tanner, and then cast Bob Saget in half of the part.
Barney Stinson isn't Jesse Katsopolis, yet both are ladies men with a
flair for the grand gesture who by their series' end learn the value
of family. Marshall Eriksen isn't Joey Gladstone, but they're both pun
spewing aspiring stand up comedians who struggle with a work-life
balance, yet ultimately believe family is the most important thing.
They're all variations on a theme - oddly specific archetypes that
together blend into the same basic friendship dynamic. Add in the
alternatingly hard-nosed, professional TV personality (Robin) and
loving, slightly lunatic mother and wife (Lily) and you can spot Becky
Donaldson too. For some time Danny dates a woman named Vicky Larson.
Sadly, their relationship doesn't work out. For Danny, family is the
most important thing. For Vicky, her career. As a News Anchor. In New
York. Just like Robin. The relationship even begins six years after he
Ted's life after becoming a widower could thus be seen as being continued by Full House:
As part of the frantic montage of Ted's key life events in the series
finale, we watch him at his dying wife's bedside, and we see him tell
his children, six years later, how he met her. What we don't see is
Ted heading home for the first time after his wife has died. We don't
see him helping his young children to move on with their lives. We
don't see, at all, his friends' reaction to the loss.
The show doesn't show that moment, though, since that moment, where a
group of friends come together to help one of them get through the
loss of his wife, is where the prequel ends, and the original series
takes back over.
Now it is hard to say if it was really that strong a connection to Full House that the creators of How I Met Your Mother intended or if it wasn't just a fitting casting choice to let the father be voiced by another famous sitcom father. But while that article might be a bit speculative (and those kinds of characters can probably be found in many sitcoms in some way or another), it gives quite some insightful arguments and some interesting connections between those two sitcoms apart from a mere voice-actor choice, even if we don't know (or might doubt) if all that was completely intentional.